3.25 \(\int \text {csch}(a+b x) \text {sech}^2(a+b x) \, dx\)

Optimal. Leaf size=23 \[ \frac {\text {sech}(a+b x)}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b} \]

[Out]

-arctanh(cosh(b*x+a))/b+sech(b*x+a)/b

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Rubi [A]  time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2622, 321, 207} \[ \frac {\text {sech}(a+b x)}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*x]*Sech[a + b*x]^2,x]

[Out]

-(ArcTanh[Cosh[a + b*x]]/b) + Sech[a + b*x]/b

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2622

Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rubi steps

\begin {align*} \int \text {csch}(a+b x) \text {sech}^2(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\text {sech}(a+b x)\right )}{b}\\ &=\frac {\text {sech}(a+b x)}{b}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\text {sech}(a+b x)\right )}{b}\\ &=-\frac {\tanh ^{-1}(\cosh (a+b x))}{b}+\frac {\text {sech}(a+b x)}{b}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 26, normalized size = 1.13 \[ \frac {\text {sech}(a+b x)}{b}+\frac {\log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[a + b*x]*Sech[a + b*x]^2,x]

[Out]

Log[Tanh[(a + b*x)/2]]/b + Sech[a + b*x]/b

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fricas [B]  time = 0.43, size = 155, normalized size = 6.74 \[ -\frac {{\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - 2 \, \cosh \left (b x + a\right ) - 2 \, \sinh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} + b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)*sech(b*x+a)^2,x, algorithm="fricas")

[Out]

-((cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 + 1)*log(cosh(b*x + a) + sinh(b*x + a) +
1) - (cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 + 1)*log(cosh(b*x + a) + sinh(b*x + a)
- 1) - 2*cosh(b*x + a) - 2*sinh(b*x + a))/(b*cosh(b*x + a)^2 + 2*b*cosh(b*x + a)*sinh(b*x + a) + b*sinh(b*x +
a)^2 + b)

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giac [B]  time = 0.14, size = 64, normalized size = 2.78 \[ \frac {\frac {4}{e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}} - \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} + 2\right ) + \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - 2\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)*sech(b*x+a)^2,x, algorithm="giac")

[Out]

1/2*(4/(e^(b*x + a) + e^(-b*x - a)) - log(e^(b*x + a) + e^(-b*x - a) + 2) + log(e^(b*x + a) + e^(-b*x - a) - 2
))/b

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maple [A]  time = 0.09, size = 23, normalized size = 1.00 \[ \frac {\frac {1}{\cosh \left (b x +a \right )}-2 \arctanh \left ({\mathrm e}^{b x +a}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(b*x+a)*sech(b*x+a)^2,x)

[Out]

1/b*(1/cosh(b*x+a)-2*arctanh(exp(b*x+a)))

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maxima [B]  time = 0.36, size = 61, normalized size = 2.65 \[ -\frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \frac {2 \, e^{\left (-b x - a\right )}}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)*sech(b*x+a)^2,x, algorithm="maxima")

[Out]

-log(e^(-b*x - a) + 1)/b + log(e^(-b*x - a) - 1)/b + 2*e^(-b*x - a)/(b*(e^(-2*b*x - 2*a) + 1))

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mupad [B]  time = 0.09, size = 52, normalized size = 2.26 \[ \frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(a + b*x)^2*sinh(a + b*x)),x)

[Out]

(2*exp(a + b*x))/(b*(exp(2*a + 2*b*x) + 1)) - (2*atan((exp(b*x)*exp(a)*(-b^2)^(1/2))/b))/(-b^2)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)*sech(b*x+a)**2,x)

[Out]

Integral(csch(a + b*x)*sech(a + b*x)**2, x)

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